The stochastic partial differential equations (REF ) with the
Dirichlet boundary condition has been studied
in [1]} and [2]}.
In the former paper, the authors used tightness arguments that conjugated
with the Skorohod theorem provided the existence of a weak stochastic
solution, in the sense that the Brownian motion, being part of the solution,
was not given in advance; while in the second one, the authors proved the
existence and uniqueness of a strong stochastic solution. Let us refer the
pioneer papers [3]} and [4]} (see also [5]}), where the
deterministic second grade equations with the Dirichlet boundary condition
were mathematically studied for the first time, and [6]} where the
deterministic equations were studied with a particular Navier boundary
condition (without friction, i.e. when \(\gamma =0\) ). The physical
interpretation of these second grade equations can be found in [7]},
[8]}, [9]}, [10]}, [11]} and [12]}. It is
relevant to recall that the deterministic methods are based on the
Faedo-Galerkin approximation method and a priori estimates. Then,
compactness arguments can be used to pass to the limit of the respective
approximate equations in the distributional sense. Unfortunately, for the
stochastic partial differential equations a priori estimates are not enough to pass
to the limit of the approximate equations, due to the lack of regularity on
the time and stochastic variables. In order to obtain a strong stochastic
solution we should verify that the sequence of the Galerkin approximations
converges strongly in some adequate topology.
